Solve for \( h \) \[ 45 h^{2}-138 h+9=0 \]

To solve the quadratic equation \( 45 h^{2} - 138 h + 9 = 0 \), we can use the quadratic formula: \[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 45 \), \( b = -138 \), and \( c = 9 \). First, we calculate the discriminant: \[ b^2 - 4ac = (-138)^2 - 4 \times 45 \times 9 \] Calculating \( (-138)^2 \): \[ (-138)^2 = 19044 \] Calculating \( 4 \times 45 \times 9 \): \[ 4 \times 45 = 180 \] \[ 180 \times 9 = 1620 \] Now substitute back into the discriminant: \[ b^2 - 4ac = 19044 - 1620 = 17424 \] Now, apply this in the quadratic formula: \[ h = \frac{138 \pm \sqrt{17424}}{90} \] Next, we need to find \( \sqrt{17424} \). Calculating this gives: \[ \sqrt{17424} \approx 131.69 \] Now substituting into the formula: \[ h = \frac{138 \pm 131.69}{90} \] Calculating the two possible values for \( h \): 1. For the positive root: \[ h = \frac{138 + 131.69}{90} = \frac{269.69}{90} \approx 2.996 \] 2. For the negative root: \[ h = \frac{138 - 131.69}{90} = \frac{6.31}{90} \approx 0.0701 \] Thus, the solutions for \( h \) are approximately: \[ h \approx 2.996 \quad \text{and} \quad h \approx 0.0701 \]